Fitting planes to early-type galaxies: MIST for the determination of the fundamental plane

The present study deals with the problem of deriving the coefficients of the fundamental plane (FP) of early-type galaxies. We introduce statistical models of the FP and relative fitting methods: the MIST (Measurement errors and Intrinsic Scatter Three dimensional) fits'. The MIST fits account for the measurement errors on the variables and their correlations as well as for the intrinsic scatter. We show that the lack of a model of the intrinsic scatter of the FP is the origin of the systematic differences between the various fitting methods. We also address the problem of estimating the uncertainties of the FP coefficients and determine a simple relation between the sample size and the expected accuracy of the coefficients. The present study leads to define a 'minimum sample size' for a correct estimate of the uncertainties. For N less than or similar to 30, both theoretical formulae and re-sampling techniques, like the bootstrap, do not give reliable estimates. The question of the 'universality' of the FP is addressed by applying the MIST fits to ten samples of cluster galaxies. The FP slopes are actually consistent between the different samples, but, due to the large uncertainties, they could also hide significant systematic differences. The feasibility of the measurement of the possible variations of the FP slopes as a function of redshift is also proved.